Sliding integral proportional (SIP) controller for aircraft skid control

ABSTRACT

The sliding, integral, and proportional controller for providing aircraft antiskid braking control includes a reference velocity subsystem, a velocity error ratio subsystem, and a main controller subsystem generating a control command output signal indicative of a command braking pressure. The main controller subsystem includes a one dimensional sliding mode controller subsystem to determine an estimated net wheel torque signal, an adaptive threshold subsystem for generating an adaptive threshold based upon the modified slip ratio signal and a clock signal, integral gain subsystems, a proportional controller subsystem, and a pressure limiter. A method for determining braking efficiency of an aircraft braking system independent of the specific conditions is also provided.

BACKGROUND OF THE INVENTION

[0001] This invention relates generally to aircraft landing gear brakingsystems, and more particularly concerns an improved system forcontrolling aircraft brake pressure.

[0002] A conventional skid detection system used in aircraft brakingsystems typically includes a wheel speed transducer for each wheel brakeof the wheels of the aircraft, for measuring wheel speed and generatingwheel speed signals that are a function of the rotational speed of thebrake wheel. The wheel speed signal is typically converted to a signalrepresenting the velocity of the aircraft, and compared with a desiredreference velocity, to generate wheel velocity error signals indicativeof the difference between the wheel velocity signals from each brakedwheel and the reference velocity signal. The output of the velocitycomparator is referred to as velocity error. The velocity error signalstypically are adjusted by a pressure bias modulator (PBM) integrator, aproportional control unit, and a compensation network, the outputs ofwhich are summed to provide an anti-skid control signal received by thecommand processor. The PBM integrator in the antiskid loop dictates themaximum allowable control pressure level during braking. When no skid isdetected, this integrator allows full system pressure to the brakes.

[0003] The conventional PID controller for aircraft brake controlsystems deals with various conditions such as aerodynamics, landing geardynamics, μ-slip profile, different landing conditions, and the like.One major problem is that tuning of controller parameters to guaranteehigh efficiency in different landing conditions and conditions affectingthe tire-runway coefficient of friction (μ) of the aircraft brakingsystem is often a difficult task.

[0004] Such algorithms usually take only one input, i.e., wheel velocity(Vw), and determine a reference velocity (Vref) with an apparatus. Thenthe Vref and Vw signals pass through the PID control logic, whichgenerates a command signal. The command signal is supplied to ahydraulic servo valve and the output of servo valve, fluid pressuregenerates a brake torque through a brake. The algorithms show goodantiskid performance—robustness and adaptability.

[0005] In spite of success of the PID type controller, related industryengineers and researchers have been continuously investigating othercontrol schemes, partially because of difficulty in antiskid brakingcontrol parameter tuning. A need therefore still exists for an antiskidbraking controller that can facilitate and shorten the process ofantiskid braking control parameter tuning. The present invention meetsthese and other needs.

SUMMARY OF THE INVENTION

[0006] Briefly, and in general terms, the present invention provides fora sliding integral proportional (SIP) controller for aircraft antiskidbraking control that improves and shortens the time required forantiskid braking control parameter tuning, and that also provides higherbraking efficiency, robustness, and adaptability, since the antiskidbraking control parameters to be tuned are adjusted based on an accurateadaptive threshold and an velocity error ratio or modified slip ratio(S_(mod)) signal with an estimated net wheel torque, a few integralgains, and a proportional gain. The proposed SIP controller requiresonly one input, and shows excellent braking efficiencies, robustness,and adaptability with only a fraction of tuning effort and time.

[0007] The present invention accordingly provides for a sliding,integral, and proportional (SIP) controller for providing anti-skidbraking control for an aircraft. The SIP controller includes a referencevelocity subsystem generating a reference velocity signal based upon aninput wheel velocity signal; a velocity error ratio subsystem generatinga modified slip ratio signal (S_(mod)) based upon a ratio of thedifference between the reference velocity and the wheel velocity to thereference velocity; and a main controller subsystem receiving thereference velocity signal and the modified slip ratio signal, andgenerating a control command output signal indicative of a commandbraking pressure.

[0008] In one embodiment, the reference velocity subsystem receives aplurality of sampled wheel velocity signals, determines a minimum valueof the sampled wheel velocity signals, and compares the minimum valuewith an individual wheel velocity signal. If the minimum value of thesampled wheel velocity signals is greater than the wheel velocitysignal, a predetermined desired reduction amount is subtracted from theminimum value of the sampled wheel velocity signals and the result isoutput as the reference velocity of the reference velocity subsystem.Otherwise the wheel velocity signal is output as the reference velocityof the reference velocity subsystem. In one aspect, the sampled wheelvelocity signals have a predetermined fixed sampling time. In a presentembodiment, the modified slip ratio signal (S_(mod)) is determined basedupon the equation: $S_{mod} = \frac{Velerror}{Vref}$

[0009] where S_(mod) is the velocity error ratio or modified slip ratio,Vref is the reference velocity in radians per second, and Velerror isthe velocity error in radians per second, determined from the equationVref−Vw, where Vw is the wheel velocity in radians per second.

[0010] In a present embodiment, the main controller subsystem includes aone dimensional sliding mode controller subsystem to determine anestimated net wheel torque signal; an adaptive threshold subsystem forgenerating an adaptive threshold based upon the modified slip ratiosignal (S_(mod)) and a clock signal; a first integral gain subsystem forcomparing the estimated net wheel torque signal with the adaptivethreshold to determine dominance between the tire drag torque andbraking torque, and outputting a corresponding gain value; a secondintegral gain subsystem exponentially generating a deep skid signal(deep_skid) when the S_(mod) signal is greater than a predeterminedlimit and a change in wheel velocity indicates a deep skid situation; athird integral gain subsystem to avoid S_(mod) signals that are toosmall or negative and to modify the initial braking command signal; aproportional controller subsystem generating an output signal to preventsudden deep skids; and a pressure limiter for limiting the commandbraking pressure. In one aspect of the invention, the output of the maincontroller subsystem is a command signal indicative of a torque, whichis converted to a command brake pressure signal by multiplication of apredetermined gain.

[0011] The estimated net wheel torque may be determined based upon thevelocity estimation error. One-dimensional sliding surface conditiontakes a form as: $\begin{matrix}{{\frac{1}{2}\frac{\quad}{t}s^{2}} = \left. {{s\frac{\partial{Vref}}{\partial t}} - \frac{{Gain}2}{{Im}w}} \middle| s \right|} & (1)\end{matrix}$

[0012] where s=Vref−{circumflex over (V)}, Vref is the referencevelocity in radians per second, {circumflex over (V)} is the observed orestimated wheel velocity in radians per second, Gain2 is determined asthe largest possible net wheel torque in ft-lbs, and Imw is thewheel/tire/brake mass moment of inertia in slug-ft². The equation (1) isalways less than zero, and thus, the sliding condition is satisfied. Thenet wheel torque signal may be determined according to the equation:$\begin{matrix}{\frac{\partial\hat{V}}{\partial t} = {\frac{{Gain}2}{{Im}w}{{sgn}(s)}}} & (2)\end{matrix}$

[0013] where sgn(s) is the sign of s. The net wheel torque signaloptionally may be determined according to the equation:

NWTe=DF×sgn(s)×Gain2   (3)

[0014] where NWTe is the estimated net wheel torque in ft-lbs, and DF isa discrete filter of time constant, 0.1 sec. The low pass filter DF maybe defined according to the equation:${DF} = \frac{0.04877}{z - 0.9512}$

[0015] where z is a complex variable.

[0016] In one embodiment of the invention, a plurality of skid levelsare established to effectively maintain a tire drag friction coefficient(μ) approaching the peak value of μ without undesirable deep skid. Inone present aspect, three skid levels are established. Thus, forexample, if the S_(mod) signal exceeds a first skid level threshold, theadaptive threshold increases to a second skid level threshold toaccommodate a braking torque and prevent a slip overshoot by apredetermined rate; if the S_(mod) signal is reduced below the secondskid level threshold, the threshold decreases to supply an appropriatebraking command and maintain the slip at the peak of μ; and the adaptivethreshold becomes a third skid level threshold greater than the secondskid level threshold and the S_(mod) signal when the runway condition isvery dry and tire drag coefficient is more than a predeterminedthreshold drag coefficient value, to generate a rapid initial brakingcommand signal. In one present aspect, if the tire drag coefficientvalue is high (more than 0.5), then the rapid initial braking commandsignal is generated for approximately 0-1.5 seconds period after brakingis initiated.

[0017] In a present embodiment, the first integral gain subsystemoutputs a first positive gain value as the integral gain output if theestimated net wheel torque is greater than or equal to the adaptivethreshold, indicating that the tire drag torque is dominant, and outputsa second negative gain value as the integral gain output if theestimated net wheel torque is less than the adaptive threshold.

[0018] In another present aspect of the invention, if S_(mod) is greaterthan the deep skid limitation (Slim) of the S_(mod) signal, and if thewheel velocity (Vw) is less than an immediately previous wheel velocity,then the deep skid signal (deep_skid) is determined according to thefollowing equation:

deep_skid=Ta3*exp(u)  (4)

[0019] where Ta3 is the first coefficient, and is a changing negativevariable determined in a look-up table based upon the referencevelocity, and u is the S_(mod) skid level determined by the followingequation: $u = \frac{\left( {S_{mod} - S_{\lim}} \right)}{0.01}$

[0020] In another aspect, the variable Ta3 changes to approximately zeroat a predetermined reference velocity, causing an increase in the brakepressure and wheel lock-up. If the wheel velocity (Vw) is greater thanor equal to an immediately previous wheel velocity when S_(mod) isgreater than or equal to Slim, then a second positive coefficient Ta3ais substituted for Ta3. In another aspect, if S_(mod) is less than aconstant value (Sneg), and if the elapsed time from the initiation ofbraking is less than about 1 second, then the output of the thirdintegral gain subsystem is a predetermined constant, multiplied by apredetermined gain. In another present aspect, if S_(mod) is less than apredetermined maximum threshold, the output signal of the proportionalcontroller subsystem is zero.

[0021] In another present embodiment, if the product of the referencevelocity (Vref) and the tire rolling radius is less than a predeterminedthreshold (Pdropout), the output signal of the proportional controllersubsystem is a predetermined constant. If the product of the referencevelocity (Vref) and the tire rolling radius is greater than or equal tothe predetermined threshold (Pdropout), then the output signal of theproportional controller subsystem is the product of the velocity errorand a predetermined negative gain. In another present aspect of theinvention, the pressure limiter limits the command braking pressurebetween about 0 and about 3000 psi.

[0022] In another present embodiment, the invention further comprises alook-up table for converting the control command output signalindicative of the command braking pressure to a control commandindicative of the command control current. In a present aspect, thelook-up table describes a nonlinear pressure vs. current relationship.In another present embodiment, the invention further comprises a currentlimiter for limiting the command control current up to about 60 mA.

[0023] The present invention also provides a method for providingsliding, integral, and proportional anti-skid braking control for anaircraft having a plurality of tires and brakes. An input wheel velocitysignal is provided, and a reference velocity signal is generated basedupon the input wheel velocity signal. A modified slip ratio signal(S_(mod)) is then generated based upon a ratio of the difference betweenthe reference velocity and the wheel velocity to the reference velocity,and a control command output signal indicative of a command brakingpressure is generated based upon the reference velocity signal and themodified slip ratio signal. In a present aspect of the method, aplurality of sampled wheel velocity signals are provided, and a minimumvalue of the sampled wheel velocity signals is determined. The minimumvalue is compared with individual wheel velocity signals, and if theminimum value of the sampled wheel velocity signals is greater than anindividual wheel velocity signal, a predetermined desired reductionamount is subtracted from the minimum value of the sampled wheelvelocity signals and the result is output as the reference velocity.Otherwise the wheel velocity signal is output as the reference velocity.In a present aspect of the method, the sampled wheel velocity signalshave a predetermined fixed sampling time.

[0024] In another aspect of the method of the invention, an estimatednet wheel torque signal is determined, an adaptive threshold isgenerated based upon the modified slip ratio signal (S_(mod)) and aclock signal, the estimated net wheel torque signal is compared with theadaptive threshold to determine dominance between the tire drag torqueand braking torque, and a corresponding first integral gain value isoutput. A deep skid signal (deep_skid) is exponentially generated whenthe S_(mod) signal is greater than a predetermined deep slid limitation(Slim) and wheel velocities indicate a deep skid situation, based uponthe modified slip ratio signal (S_(mod)), the wheel velocity signal(Vw), the reference velocity signal (Vref), the tire rolling radius, apredetermined deep skid limitation (Slim) of the S_(mod) signal, andfirst and second function coefficients. The initial braking commandsignal is modified to avoid S_(mod) signals that are too small ornegative; an output signal is generated to prevent sudden deep skids;and the command braking pressure is limited to a maximum amount.

[0025] The present invention also provides for a method for determiningbraking efficiency of an aircraft braking system independent of thespecific conditions. A new μefficiency (η) is determined based upon anantiskid braking efficiency (μ_(b)), average braking force (A) of allthe non-braking forces acting to stop, or accelerate the aircraft, andthe average braking force (B) of the aircraft braking system, accordingto the following equation: $\begin{matrix}{\eta = \frac{A + {\mu_{b} \cdot B}}{A + B}} & (6)\end{matrix}$

[0026] where A is the force of all the non-braking forces acting tostop, or accelerate the aircraft; B is the force of the aircraft brakingsystem, and μ_(b) is the antiskid braking efficiency, determined as theactual tire drag coefficient μ divided by the peak tire drag coefficientμ.

[0027] These and other aspects and advantages of the invention willbecome apparent from the following detailed description and theaccompanying drawings, which illustrate by way of example the featuresof the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0028]FIG. 1 is a schematic diagram of a sliding integral proportional(SIP) controller according to the present invention.

[0029]FIG. 2 is a schematic diagram of the velocity error ratiosubsystem of the controller of FIG. 1.

[0030]FIG. 3 is a schematic diagram of the main controller subsystem ofthe controller of FIG. 1.

[0031]FIG. 4 is a schematic diagram of the sliding mode controllersubsystem of the controller of FIG. 1.

[0032]FIG. 5 is a schematic diagram of the first integrator of thecontroller of FIG. 1.

[0033]FIG. 6 is a schematic diagram of the second integrator of thecontroller of FIG. 1.

[0034]FIG. 7 is a schematic diagram of the third integrator of thecontroller of FIG. 1.

[0035]FIG. 8 is a schematic diagram of the proportional controllersubsystem of the controller of FIG. 1.

[0036]FIG. 9A is a graph of the Mu (μ) vs. Slip Velocity for theaircraft B717 medium landing and μ=0.5.

[0037]FIG. 9B is a graph of the Mu (μ) vs. Slip ratio for the aircraftB717 medium landing and μ=0.5.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0038] While aircraft brake control systems typically deal with variousconditions such as aerodynamics, landing gear dynamics, μ-slip profile,different landing conditions, and the like, a major problem with the useof such controllers has been appropriate tuning of parameters for thecontroller to provide for efficient operation of such controllers indifferent landing conditions that affect the tire-runway coefficient offriction (μ) of the aircraft braking system.

[0039] As is illustrated in the drawings, the invention is embodied in asliding, integral, and proportional (SIP) controller for aircraftantiskid brake control systems that utilizes a one dimensional slidingcontroller combined with an adaptive threshold subsystem, integralgains, and a proportional gain for providing anti-skid braking control.A reference velocity signal is used as an input of a sliding modecontroller-subsystem to estimate a net wheel torque signal. A modifiedslip ratio signal (S_(mod)) is generated by a velocity error ratiosubsystem, and a deep skid signal is generated exponentially when theS_(mod) signal is greater than a given limit and wheel velocitiesindicate a deep skid situation.

[0040] The controller of the invention has been developed based onone-dimensional sliding control theory (Slotine et al., 1991) andcombined with an adaptive threshold subsystem, a few integral gains anda proportional controller.

[0041] Referring to FIG. 1, the SIP controller 20 receives only oneexternal input, the wheel velocity (Vw), typically provided by a wheelspeed transducer 22 operatively connected to a wheel of an aircraft (notshown), and the reference velocity (Vref) is calculated in a referencevelocity subsystem 24 based on the wheel velocity. The referencevelocity subsystem unit takes, for example, 10 sampled signals with anappropriate fixed sampling time from the input-wheel velocity signal anda minimum value is chosen. This minimum value is compared with a wheelvelocity signal. If the value is greater than the wheel velocity signal,an appropriate reduction amount is subtracted and the result is used asa reference velocity; otherwise, the wheel velocity goes to the outputof the subsystem unit.

[0042] The wheel velocity (Vw) is subtracted from the reference velocityat a summing junction 26, resulting in a velocity error signal 28(Velerror) which is received by a velocity error ratio subsystem 30. Thevelocity error (Velerror) is measured in radians per second, and isdetermined according to the equation:

Velocity Error (rad/sec)=Vref−Vw

[0043] As is illustrated in FIG. 2, the velocity error ratio subsystemgenerates a velocity error ratio, also referred to as the modified slipratio (S_(mod)), according to the following equation:$S_{mod} = \frac{Velerror}{Vref}$

[0044] Since the aircraft velocity signal (Vac) is not measured in thepresent antiskid braking system, accurate slip (Vac−Vw) or slip ratio(Vac−Vw)/Vac are not utilized. Thus, the velocity error ratio ormodified slip ratio (S_(mod)) is calculated to obtain the adaptivethreshold and integral gains, instead.

[0045] In addition to the reference velocity subsystem and velocityerror ratio subsystem, as is illustrated in FIG. 1, the SIP controllerincludes a main controller subsystem 32, a look-up table #1 (34), and acurrent limiter 36 yielding the final braking command current i_(com).The S_(mod), Vref, Vw and Velerror signals are the major inputs of themain controller subsystem, the output of which is the control commandcurrent i_(com). The command pressure, P_(com) is converted to i_(com)through the look-up table #1, which describes a nonlinear pressure vs.current relationship.

[0046] Referring to FIG. 3, the main controller comprises a sliding modecontroller subsystem 38, an adaptive threshold subsystem 40, first,second and third integrator subsystems 42, 44, 46, a proportionalcontroller subsystem 48, and a pressure limiter 50. The outputs of thethree integrator subsystems are summed at a first summer 52 over aperiod defined by $\frac{Ts}{z - 1},$

[0047] where Ts is a control cycle time, and z is a complex variable.The output of the proportional controller is summed with the output ofthe first summer at a second summer 54, yielding an output of the maincontroller subsystem. The output of the main controller subsystem isdimensionally torque, and is converted to a pressure signal 56 bymultiplication of Gain1 58, after which the output pressure signal(P_(com)) is limited by the pressure limiter between 0 and 3000 psi.

[0048] Sliding Mode Controller

[0049] Referring to FIGS. 3 and 4, the reference velocity (rad/sec)signal is received by the sliding mode controller subsystem as an inputfor generating a corresponding net wheel torque (NWTe) signal (inft-lbs.) input to the first integrator subsystem.

[0050] To design a successfully stable sliding mode controller,estimation error vectors need to slide towards zero as quickly aspossible along a series of hyper plane intersections (Slotine et al.,1991). In this controller, a one-dimensional sliding surface conditionis met, because only one estimation error (Ve) is obtained. Theestimated net wheel torque may be determined based upon the velocityestimation error, i.e., Ve=Vref−{circumflex over (V)}

[0051] One dimensional sliding surface condition is expressed as:

[0052] $\begin{matrix}{{\frac{1}{2}\frac{\quad}{t}s^{2}} = \left. {{s\frac{\partial{Vref}}{\partial t}} - \frac{{Gain}2}{{Im}w}} \middle| s \right|} & (1)\end{matrix}$

[0053] where s=Vref−V, Vref is the reference velocity in radians persecond, {circumflex over (V)} is the observed or estimated wheelvelocity in radians per second, Gain2 is determined as the largestpossible net wheel torque in ft-lbs, and Imw is the wheel/tire/brakemass moment of inertia in slug-ft².

[0054] Referring to FIG. 4, s is determined in a third summer 60 as thedifference between the reference velocity and the observed or estimatedwheel velocity, and s is input to block 62, the output of which isamplified by Gain2 at 64. The factor Gain2 is dimensionally a torque,and the equation (1) is always less than zero. Therefore, the slidingcondition is satisfied. By assessing Gain2 as the largest possibletorque, the wheel dynamics for estimating wheel velocity and NWTe may beexpressed as $\begin{matrix}{\frac{\partial\hat{V}}{\partial t} = {\frac{{Gain}2}{{Im}w}{{sgn}(s)}}} & (2)\end{matrix}$

[0055] where sgn(s) is the sign of s.

[0056] A discrete filter (DF) 66 of time constant, 0.1 sec, receivingthe output of Gain2 and defined, for example, by a function such as${{DF} = \frac{0.04877}{z - 0.9512}},$

[0057] may be used as a low pass filtering means, according to theequation:

NWTe=DF×sgn(s)×Gain2   (3)

[0058] where NWTe is the estimated net wheel torque in ft-lbs, and DF isa discrete filter of time constant, 0.1 sec. The estimated net wheeltorque signal is received as feedback by amplifier 68, and integratedover a period defined by block 70 to provide the observed or estimatedwheel velocity {circumflex over (V)} to summer 60.

[0059] Adaptive Threshold

[0060] Referring to FIG. 3, the modified slip ratio signal (S_(mod)) anda clock signal are injected as two inputs to the adaptive thresholdsubsystem, that generates an adaptive threshold as an input to the firstintegrator subsystem.

[0061] To effectively maintain a tire drag friction coefficient (μ) atthe vicinity of the peak value of μ without undesirable deep skid, threeskid levels are established such as skid-level 1, skid-level 2, andskid-level 3 in the μ vs. S_(mod) configuration. If a S_(mod) signalreaches skid-level 1 and exceeds the level, the threshold increases toaccommodate a braking torque and prevent a slip overshoot by apredetermined rate. If the S_(mod) signal is reduced below skid-level 2,the threshold decreases to supply an appropriate braking command andmaintain the slip at the peak of μ.

[0062] To generate a rapid initial braking command signal, such as whenthe runway condition is very dry and tire drag coefficient is high (forexample, more than 0.5), the skid-level 3 is established. The levelneeds to be a little higher than the other two skid levels and S_(mod)signal for the initial 0-1.5 seconds period after braking is initiated.

[0063] The First Integral Gain Subsystem

[0064] Referring to FIGS. 3 and 5, the output of sliding modecontroller, NWTe, is compared with the adaptive threshold in the firstintegral gain subsystem 42 to evaluate dominance between the tire dragtorque and braking torque. Once the dominance is evaluated, acorresponding gain value (T_(apply)) is determined as the output of thesubsystem. For example, if NWTe is greater than the adaptive threshold,that is, the tire drag torque is dominant, then a positive gain valueTa1 goes to output of the subsystem; if NWTe is less than the adaptivethreshold, in other words the braking torque is being appliedexcessively, then a negative gain value Ta2 is supplied as an integralgain.

[0065] Although only one of two gains is determined in the subsystemthrough the evaluation of torque dominance using the adaptive threshold,the system operates to modulate a pressure bias.

[0066] The Second Integral Gain Subsystem

[0067] Referring to FIGS. 3 and 6, the inputs of the second integralgain subsystem are S_(mod), Vw, Vref, r_r (the tire rolling radius inft), Slim (the deep skid limitation in μ vs. S_(mod) configuration), afirst coefficient Ta3, and a second coefficient Ta3 a. The output is adeep skid signal (deep_skid).

[0068] Referring to FIG. 9B, when the abscissa of vertex of accurate μvs. slip ratio is 0.056, the corresponding S_(mod) limitation may bechosen as 0.038 and this skid limitation is termed as Slim here. If theS_(mod) is greater than this limitation and Vw is less than Vw_prv(previous Vw), then the deep skid signal is released by the exponentialfunction #1.

Function #1: deep_skid=Ta3*exp(u)  (4)

[0069] where $u = \frac{\left( {{S{mod}} - {S\lim}} \right)}{0.01}$

[0070] The deep skid (DS) signal amplifies exponentially depending onthe S_(mod) skid level, u. Furthermore, Ta3 is not a constant but achanging negative variable along the reference velocity, which isdetermined in a look-up table #2 72. The coefficient Ta3 is a variablethat changes to approximately zero at a certain Vref level (P dropout),i.e., Ta3=0, if Vref≦P_dropout. This causes the brake pressure increaseand wheel lock-up, when the Vref reaches the P_dropout.

[0071] In anti-skid control, the wheel velocity is often quite close tothe aircraft velocity immediately after a large decrease in brakepressure, which may result in a low braking efficiency. To remedy this,the wheel velocity (Vw) is monitored in discrete intervals, with thewheel velocity (Vw) of a given interval being compared with a wheelvelocity Vw_prv of an immediately previous interval, under the conditionof S_(mod)≧Slim. If Vw is greater than or equal to Vw_prv, then adifferent positive coefficient Ta3a is incorporated into the exponentialfunction #2.

Function #2: deep_skid=Ta3a*exp(u)  (5)

[0072] The Third Integral Gain Subsystem

[0073] Referring to FIGS. 3 and 7, to avoid S_(mod) signals that are toosmall or negative, a simple logic is utilized in the third integral gainsubsystem, illustrated in the flow diagram of FIG. 7. If S_(mod) is lessthan a constant value (Sneg), then the output skid_neg=Ta0. Ta0 is apredetermined constant. To enhance the initial braking command signalwithin about 0 to 1.0 second, one device is needed besides theadjustment of skid_level3 of the adaptive threshold subsystem, that is,the constant Ta0 is multiplied by a gain (enhance_init), and thus anoutput signal (Skid_neg) is generated by the third integral gainsubsystem.

[0074] The Proportional Controller

[0075] Referring to the flow chart illustrated in FIG. 8, a proportionalcontroller 48 is incorporated into the SIP controller to prevent suddendeep skids, which are often observed in aircraft antiskid brakingsimulations and flight test data, even under constant μ runwayconditions.

[0076] One feature of the proportional controller that is different froma conventional proportional controller is that the proportional controlaction occurs only when the Vref is above a certain level (Pdropout). Ifthe Vref reaches Pdropout and proceeds below it, then the output changesto a constant value (T_p0), and this also helps a necessary rapid wheellock-up near dead stop as the coefficient Ta3 drops to zero in the thirdintegral gain subsystem.

[0077] FIGS. 9A, and 9B illustrate some antiskid physics obtained forthe aircraft B717 medium landing and μ=0.5. FIG. 9A demonstrates thatthe tire-runway coefficient of friction (μ) changes as the aircraft slipvelocity decreases. The peak μ is located around the slip velocity of12.5 ft/sec. However, from FIG. 9B it can be seen that the μ vs. slipratio curve does not vary significantly during the braking and aircraftdeceleration. From FIGS. 9A and 9B, it may be observed that the SIPcontroller manages antiskid performance near the peak μ (0.5) or frontside of the curves well. In FIG. 9B, the peak μ is found at the slipratio of about 0.056.

[0078] For the main landing gear dynamic model, two D.O.F. nonlinearlagrangian equations based on a T-shape gear configuration areincorporated in the Simulation model. The Simulation shows a closecorrelation between the brake torque and gear walk stability. It isnoted that sudden drops of brake torque amplifies slightly the gear walkvelocity.

[0079] The SIP controller of the invention has been tested for broadrange of tire-runway friction coefficients for B717 parameters in a nonreal time environment. The simulation results show high efficiency, i.e,97.4 % for Mumax=0.1, 94.3% for Mumax 0.5, 92.4% for Mu-step 0.4-0.2.The dynamic model of aircraft, wheel, main and nose landing gears,hydraulic system, torque data, and aerodynamic models have beencarefully examined and incorporated in the closed loop control model totest the SIP controller and to evaluate antiskid performance.

[0080] The main landing gear stability has been also tested with anexternal pulse force by the assumption that fore-aft gear dampingratio=0 and up to −15% of critical damping. Very excellent dampingeffects have been observed even for −15% damping ratio under thecondition that a horizontal axle acceleration would be available andadded to the control signal output.

[0081] The simulations have been performed for the B717 aircraftparameters, T-shape main landing gears, a hydraulic system, and brakesin a non real time environment.

[0082] The simulation results show a successful performance with highefficiencies for various tire drag friction coefficients (μ) with only afew tuning parameters. TABLE 1 Braking Efficiency, B717 Medium Landing μEfficiency* 0.1 97.4% 0.5 94.3% Stepped μ 0.4-0.2 92.4%

[0083] In table 1, the Efficiency* is a higher efficiency by comparisonamong the cumulative distance efficiency and new μ efficiency, explainedbelow.

[0084] Braking Performance

[0085] The New μ Efficiency Method:

[0086] The new μ efficiency (η) of the present invention, and cumulativedistance efficiency (described in the Hydro-Aire Default Simulation PlanR 1397) are interrelated for any set of conditions. $\begin{matrix}{\eta = \frac{A + {\mu_{b} \cdot B}}{A + B}} & (6)\end{matrix}$

[0087] where A is the average braking force of all the non-braking (i.e.aerodynamic) forces that are acting to stop, or accelerate the aircraft,the main factors of which are flaps, spoilers, body drag, thrustreversers, wind, runway slope, and the like; B is the average brakingforce assuming the braking is perfect; and μ_(b) is the antiskid brakingefficiency, actual μ divided by the peak μ.

[0088] In this way, the braking performance is calculated independent ofthe specific conditions. The new μ efficiency method shows a lowerefficiency as the overall efficiency decreases, especially for theμ-step tire runway condition. However, the higher the efficiency, thesmaller the difference.

[0089] Gear Stability with a Horizontal Component of Wheel AxleAcceleration:

[0090] It should be appreciated that if a wheel axle acceleration signalis available in future, the horizontal component of accelerationmultiplied by a gain can be directly added to the control current outputsignal, which can increase the overall closed loop damping effect of thesystem dramatically. Excellent gear stability is expected even fornegative value of fore-aft landing gear damping up to 15% of thecritical damping ratio.

[0091] It will be apparent from the foregoing that while particularforms of the invention have been illustrated and described, variousmodifications can be made without departing from the spirit and scope ofthe invention. Accordingly, it is not intended that the invention belimited, except as by the appended claims.

What is claimed is:
 1. A sliding, integral, and proportional controllerfor providing anti-skid braking control for an aircraft having aplurality of tires and brakes, comprising: a reference velocitysubsystem generating a reference velocity signal based upon an inputwheel velocity signal; a velocity error ratio subsystem generating amodified slip ratio signal (S_(mod)) based upon a ratio of thedifference between the reference velocity and the wheel velocity to thereference velocity; and a main controller subsystem receiving thereference velocity signal and the modified slip ratio signal, the maincontroller subsystem generating a control command output signalindicative of a command braking pressure responsive to the referencevelocity signal and the modified slip ratio signal.
 2. The sliding,integral, and proportional controller of claim 1, wherein the referencevelocity subsystem receives a plurality of sampled wheel velocitysignals and determines a minimum value of the sampled wheel velocitysignals, compares the minimum value with an individual wheel velocitysignal, and wherein if the minimum value of the sampled wheel velocitysignals is greater than the wheel velocity signal, a predetermineddesired reduction amount is subtracted from the minimum value of thesampled wheel velocity signals and the result is output as the referencevelocity of the reference velocity subsystem, and otherwise the wheelvelocity signal is output as the reference velocity of the referencevelocity subsystem.
 3. The sliding, integral, and proportionalcontroller of claim 2, wherein the sampled wheel velocity signals have apredetermined fixed sampling time.
 4. The sliding, integral, andproportional controller of claim 1, wherein the modified slip ratiosignal (S_(mod)) is determined based upon the equation:$S_{mod} = \frac{Velerror}{Vref}$

where S_(mod) is the velocity error ratio or modified slip ratio, Vrefis the reference velocity in radians per second, and Velerror is thevelocity error in radians per second, determined from the equationVref−Vw, where Vw is the wheel velocity in radians per second.
 5. Thesliding, integral, and proportional controller of claim 1, wherein themain controller subsystem comprises: a one dimensional sliding modecontroller subsystem to determine an estimated net wheel torque signal;an adaptive threshold subsystem for generating an adaptive thresholdbased upon the modified slip ratio signal (S_(mod)) and a clock signal;a first integral gain subsystem for comparing the estimated net wheeltorque signal with the adaptive threshold to determine dominance betweenthe tire drag torque and braking torque, and outputting a correspondinggain value; a second integral gain subsystem exponentially generating adeep skid signal (DS) when the S_(mod) is greater than a predeterminedlimit and wheel velocities indicating a deep skid situation based uponthe modified slip ratio signal (S_(mod)), the wheel velocity signal(Vw), the reference velocity signal (Vref), the tire rolling radius, apredetermined deep skid limitation (Slim) of the S_(mod) signal, andfirst and second function coefficients; a third integral gain subsystemto modify the initial braking command signal to avoid S_(mod) signalsthat are too small or negative; a proportional controller subsystemgenerating an output signal to prevent sudden deep skids; and a pressurelimiter for limiting the command braking pressure.
 6. The sliding,integral, and proportional controller of claim 5, wherein the output ofthe main controller subsystem is a torque.
 7. The sliding, integral, andproportional controller of claim 6, wherein the torque is converted to acommand brake pressure signal by multiplication of a predetermined gain.8. The sliding, integral, and proportional controller of claim 5,wherein the velocity estimation error (Ve) is determined from theequation: $\begin{matrix}{{\frac{1}{2}\frac{\quad}{t}s^{2}} = \left. {{s\frac{\partial{Vref}}{\partial t}} - \frac{{Gain}2}{{Im}w}} \middle| s \right|} & (1)\end{matrix}$

where Vref is the reference velocity in radians per second, {circumflexover (V)} is the observed or estimated wheel velocity in radians persecond, s=Vref−{circumflex over (V)}, Gain2 is determined as the largestpossible net wheel torque in ft-lbs, and Imw is the wheel/tire/brakemass moment of inertia in slug-ft².
 9. The sliding, integral, andproportional controller of claim 5, wherein the largest possible torqueis determined as the Gain2, and the net wheel torque signal isdetermined according to the equation: $\begin{matrix}{\frac{\partial\hat{V}}{\partial t} = {\frac{{Gain}2}{{Im}w}{{sgn}(s)}}} & (2)\end{matrix}$

where Gain2 is determined as the largest possible net wheel torque inft-lbs, and Imw is the wheel/tire/brake mass moment of inertia inslug-ft².
 10. The sliding, integral, and proportional controller ofclaim 5, wherein the net wheel torque signal is determined according tothe equation: NWTe=DF×sgn(s)×Gain2   (3)where NWTe is the estimated netwheel torque in ft-lbs, DF is a discrete filter of time constant, 0.1sec, sgn(s) is the sign of s, and Gain 2 is determined as the largestpossible net wheel torque in ft-lbs.
 11. The sliding, integral, andproportional controller of claim 5, wherein DF is a low pass filterdefined according to the equation ${DF} = \frac{0.04877}{z - 0.9512}$

where DF is a discrete filter of time constant, 0.1 sec.
 12. Thesliding, integral, and proportional controller of claim 5, wherein aplurality of skid levels are established to effectively maintain a tiredrag friction coefficient (μ) at the vicinity of the peak value of μwithout undesirable deep skid.
 13. The sliding, integral, andproportional controller of claim 12, wherein three skid levels areestablished.
 14. The sliding, integral, and proportional controller ofclaim 12, wherein if the S_(mod) signal exceeds a first skid levelthreshold, the adaptive threshold increases to a second skid levelthreshold to accommodate a braking torque and prevent a slip overshootby a predetermined rate; if the S_(mod) signal is reduced below thesecond skid level threshold, the threshold decreases to supply anappropriate braking command and maintain the slip at the peak of μ; andthe adaptive threshold becomes a third skid level threshold greater thanthe second skid level threshold and the S_(mod) signal when the runwaycondition is very dry and tire drag coefficient is more than apredetermined threshold drag coefficient value, to generate a rapidinitial braking command signal.
 15. The sliding, integral, andproportional controller of claim 14, wherein the predetermined thresholddrag coefficient value is about 0.5.
 16. The sliding, integral, andproportional controller of claim 14, wherein the rapid initial brakingcommand signal is generated for the approximately an initial 0-1.5seconds period after braking is initiated.
 17. The sliding, integral,and proportional controller of claim 5, wherein the first integral gainsubsystem outputs a first positive gain value as an integral gain if theestimated net wheel torque is greater than or equal to the adaptivethreshold, indicating that the tire drag torque is dominant, andoutputting a second negative gain value as an integral gain if theestimated net wheel torque is less than the adaptive threshold.
 18. Thesliding, integral, and proportional controller of claim 5, wherein thedeep skid limitation (Slim) of the S_(mod) signal is approximately0.038.
 19. The sliding, integral, and proportional controller of claim5, wherein if S_(mod) is greater than the deep skid limitation (Slim) ofthe S_(mod) signal, and if the wheel velocity (Vw) is less than animmediately previous wheel velocity, then the deep skid signal isdetermined according to the following equation: deep _(—)skid=Ta3*exp(u)  (4)where u is the S_(mod) skid level determined by thefollowing equation:$u = \frac{\left( {{S{mod}} - {S\lim}} \right)}{0.01}$

where Ta3 is the first coefficient, and is a changing negative variabledetermined in a look-up table based upon the reference velocity.
 20. Thesliding, integral, and proportional controller of claim 19, wherein thevariable Ta3 changes to approximately zero at a predetermined referencevelocity, causing an increase in the brake pressure and wheel lock-up;and wherein if the wheel velocity (Vw) is greater than or equal to animmediately previous wheel velocity when S_(mod) is greater than orequal to Slim, then a second positive coefficient Ta3 a is substitutedfor Ta3.
 21. The sliding, integral, and proportional controller of claim5, wherein if S_(mod) is less than a constant value (Sneg), and if theelapsed time from the initiation of braking is less than about 1 second,then the output of the third integral gain subsystem is a predeterminedconstant, multiplied by a predetermined gain.
 22. The sliding, integral,and proportional controller of claim 5, wherein if S_(mod) is less thana predetermined maximum threshold, the output signal of the proportionalcontroller subsystem is zero.
 23. The sliding, integral, andproportional controller of claim 22, wherein if the product of thereference velocity (Vref) and the tire rolling radius is less than apredetermined threshold, the output signal of the proportionalcontroller subsystem is a predetermined constant; and wherein if theproduct of the reference velocity (Vref) and the tire rolling radius isgreater than or equal to a predetermined threshold, then the outputsignal of the proportional controller subsystem is the product of thevelocity error and a predetermined negative gain.
 24. The sliding,integral, and proportional controller of claim 5, wherein the pressurelimiter limits the command braking pressure between about 0 and about3000 psi.
 25. The sliding, integral, and proportional controller ofclaim 1, further comprising a look-up table for converting the controlcommand output signal indicative of the command braking pressure to acommand control indicative of the command current.
 26. The sliding,integral, and proportional controller of claim 25, wherein the look-uptable describes a nonlinear pressure vs. current relationship.
 27. Thesliding, integral, and proportional controller of claim 25, furthercomprising a current limiter for limiting the control command signalcurrent up to about 60 mA.
 28. A method for determining brakingefficiency of an aircraft braking system independent of the specificconditions, comprising determining the new μ efficiency (η) based uponan antiskid braking efficiency (μ_(b)), average braking force (A) of allthe non-braking forces acting to stop, or accelerate the aircraft, andthe average braking force (B) of the aircraft braking system, accordingto the following equation: $\begin{matrix}{\eta = \frac{A + {\mu_{b} \cdot B}}{A + B}} & (6)\end{matrix}$

where A is the of all the non-braking forces acting to stop, oraccelerate the aircraft; B is the of the aircraft braking system, andμ_(b) is the antiskid braking efficiency, determined as the actual tiredrag coefficient μ divided by the peak tire drag coefficient μ.
 29. Thesliding, integral and proportional controller of claim 1, wherein asignal proportional to a horizontal component of acceleration ismultiplied by a predetermined gain signal, and the resulting value isadded to the control command output signal.
 30. A method for providingsliding, integral, and proportional anti-skid braking control for anaircraft having a plurality of tires and brakes, comprising: providingan input wheel velocity signal; generating a reference velocity signalbased upon the input wheel velocity signal; generating a modified slipratio signal (S_(mod)) based upon a ratio of the difference between thereference velocity and the wheel velocity to the reference velocity; andgenerating a control command output signal indicative of a commandbraking pressure responsive to the reference velocity signal and themodified slip ratio signal.
 31. The method of claim 30, whereinproviding an input wheel velocity signal comprises providing a pluralityof sampled wheel velocity signals, and determining a minimum value ofthe sampled wheel velocity signals, and further comprising: comparingthe minimum value with an individual wheel velocity signal, and whereinif the minimum value of the sampled wheel velocity signals is greaterthan the wheel velocity signal, a predetermined desired reduction amountis subtracted from the minimum value of the sampled wheel velocitysignals and outputting the result as the reference velocity, andotherwise outputting the wheel velocity signal as the referencevelocity.
 32. The method of claim 31, wherein the sampled wheel velocitysignals have a predetermined fixed sampling time.
 33. The method ofclaim 30, wherein the modified slip ratio signal (S_(mod)) is determinedbased upon the equation: $S_{mod} = \frac{Velerror}{Vref}$

where S_(mod) is the velocity error ratio or modified slip ratio, Vrefis the reference velocity in radians per second, and Velerror is thevelocity error in radians per second, determined from the equationVref−Vw, where Vw is the wheel velocity in radians per second.
 34. Themethod of claim 30, wherein generating a control command output signalcomprises: determining an estimated net wheel torque signal; generatingan adaptive threshold based upon the modified slip ratio signal(S_(mod)) and a clock signal; comparing the estimated net wheel torquesignal with the adaptive threshold to determine dominance between thetire drag torque and braking torque, and outputting a correspondingfirst integral gain value; exponentially generating a deep skid signal(deep_skid) when the S_(mod) is greater than a predetermined limit andwheel velocities indicating a deep skid situation based upon themodified slip ratio signal (S_(mod)), the wheel velocity signal (Vw),the reference velocity signal (Vref), the tire rolling radius, apredetermined deep skid limitation (Slim) of the S_(mod) signal, andfirst and second function coefficients; modifying the initial brakingcommand signal to avoid S_(mod) signals that are too small or negative;generating an output signal to prevent sudden deep skids; and limitingthe command braking pressure.
 35. The method of claim 34, wherein thecontrol command output signal is a torque.
 36. The method of claim 35,wherein the torque is converted to a command brake pressure signal bymultiplication of a predetermined gain.
 37. The method of claim 34,wherein the velocity estimation error (Ve) is determined from theequation: $\begin{matrix}{{\frac{1}{2}\frac{}{t}s^{2}} = {{s\frac{\partial{Vref}}{\partial t}} - {\frac{{Gain}2}{{Im}w}{s}}}} & (1)\end{matrix}$

where Vref is the reference velocity in radians per second, {circumflexover (V)} is the observed or estimated wheel velocity in radians persecond, s=Vref−V, Gain2 is determined as the largest possible net wheeltorque in ft-lbs, and Imw is the wheel/tire/brake mass moment of inertiain slug-ft².
 38. The method of claim 34, wherein the largest possibletorque is determined as the Gain2, and the net wheel torque signal isdetermined according to the equation: $\begin{matrix}{\frac{\partial\hat{V}}{\partial t} = {\frac{{Gain}2}{{Im}w}{{sgn}(s)}}} & (2)\end{matrix}$

where Gain2 is determined as the largest possible net wheel torque inft-lbs, and Imw is the wheel/tire/brake mass moment of inertia inslug-ft².
 39. The method of claim 34, wherein the net wheel torquesignal is determined according to the equation: NWTe=DF×sgn(s)×Gain2  (3)where NWTe is the estimated net wheel torque in ft-lbs, DF is adiscrete filter of time constant, 0.1 sec, sgn(s) is the signal s, andGain 2 is determined as the largest possible net wheel torque in ft-lbs.40. The method of claim 34, wherein DF is a low pass filter definedaccording to the equation ${DF} = \frac{0.04877}{z - 0.9512}$

where DF is a discrete filter of time constant, 0.1 sec.
 41. The methodof claim 34, wherein a plurality of skid levels are established toeffectively maintain a tire drag friction coefficient (μ) at thevicinity of the peak value of u without undesirable deep skid.
 42. Themethod of claim 41, wherein three skid levels are established.
 43. Themethod of claim 41, wherein if the S_(mod) signal exceeds a first skidlevel threshold, the adaptive threshold increases to a second skid levelthreshold to accommodate a braking torque and prevent a slip overshootby a predetermined rate; if the S_(mod) signal is reduced below thesecond skid level threshold, the threshold decreases to supply anappropriate braking command and maintain the slip at the peak of μ; andthe adaptive threshold becomes a third skid level threshold greater thanthe second skid level threshold and the S_(mod) signal when the runwaycondition is very dry and tire drag coefficient is more than apredetermined threshold drag coefficient value, to generate a rapidinitial braking command signal.
 44. The method of claim 43, wherein thepredetermined threshold drag coefficient value is about 0.5.
 45. Themethod of claim 43, wherein the rapid initial braking command signal isgenerated for the approximately an initial 0-1.5 seconds period afterbraking is initiated.
 46. The method of claim 34, wherein outputting thefirst integral gain comprises outputting a first positive gain value ifthe estimated net wheel torque is greater than or equal to the adaptivethreshold, indicating that the tire drag torque is dominant, andoutputting a second negative gain value if the estimated net wheeltorque is less than the adaptive threshold.
 47. The method of claim 34,wherein the deep skid limitation (Slim) of the S_(mod) signal isapproximately 0.038.
 48. The method of claim 34, wherein if S_(mod) isgreater than the deep skid limitation (Slim) of the S_(mod) signal, andif the wheel velocity (Vw) is less than an immediately previous wheelvelocity, then the deep skid signal is determined according to thefollowing equation: DS=Ta3*exp(u)  (4)where u is the S_(mod) skid leveldetermined by the following equation:$u = \frac{\left( {{Smod} - {Slim}} \right)}{0.01}$

where Ta3 is the first coefficient, and is a changing negative variabledetermined in a look-up table based upon the reference velocity.
 49. Themethod of claim 48, wherein the variable Ta3 changes to approximatelyzero at a predetermined reference velocity, causing an increase in thebrake pressure and wheel lock-up; and wherein if the wheel velocity (Vw)is greater than or equal to an immediately previous wheel velocity whenS_(mod) is greater than or equal to Slim, then a second positivecoefficient Ta3 a is substituted for Ta3.
 50. The method of claim 34,wherein if S_(mod) is less than a constant value (Sneg), and if theelapsed time from the initiation of braking is less than about 1 second,then the output of the third integral gain subsystem is a predeterminedconstant, multiplied by a predetermined gain.
 51. The method of claim34, wherein if S_(mod) is less than a predetermined maximum threshold,the output signal of the proportional controller subsystem is zero. 52.The method of claim 51, wherein if the product of the reference velocity(Vref) and the tire rolling radius is less than a predeterminedthreshold, the output signal of the proportional controller subsystem isa predetermined constant; and wherein if the product of the referencevelocity (Vref) and the tire rolling radius is greater than or equal toa predetermined threshold, then the output signal of the proportionalcontroller subsystem is the product of the velocity error and apredetermined negative gain.
 53. The method of claim 34, wherein thepressure limiter limits the command braking pressure between about 0 andabout 3000 psi.
 54. The method of claim 31, further comprisingconverting the control command output signal indicative of the commandbraking pressure to a command control current indicative of the commandbraking pressure according to a look-up table.
 55. The method of claim54, wherein the look-up table describes a nonlinear pressure vs. currentrelationship.
 56. The method of claim 54, further comprising limitingthe control command signal current up to about 60 mA.
 57. The method ofclaim 30, further comprising multiplying a signal representing thehorizontal component of acceleration of a wheel by a predetermined gainnumber and adding the resulting value to the control current outputsignal to increase closed loop damping